M. Gómez-Gesteira

SPATIOTEMPORAL STRUCTURES IN DISCRETELY-COUPLED ARRAYS OF NONLINEAR CIRCUITS: A REVIEW

Spatiotemporal pattern formation occurring in discretely-coupled nonlinear dynamical systems has been studied numerically. Reaction-diffusion systems can be viewed as an assembly of a large number of identical local subsystems which are coupled to each other by diffusion. Here, the local subsystems are defined by a system of nonlinear ordinary differential equations. While for continuous systems, the characteristic time scale corresponding to the diffusion is slower than that corresponding to the local subsystems, in discretely-coupled systems, both time scales can be of the same order of magnitude. Discrete systems can exhibit behaviors different from those exhibited by their equivalent continuous model: the wave propagation failure phenomenon occurring in nerve-pulse propagation due to transmission blockage is a case in point. In this case, it is found that the wave fails to propagate at or below some critical value of the coupling coefficient. Systems of coupled cells can be found to occur in the trans...

Vulnerability in excitable Belousov-Zhabotinsky medium: from 1D to 2D

Abstract Mechanisms for initiating rotating waves in 1D and 2D excitable media were compared and parameters affecting wavefront formation were analyzed. The time delay between two sequentially initiated wavefronts (a conditioning wave followed by a test wave) was varied in order to induce rotating waves, a protocol similar to that utilized in cardiac muscle experiments to reveal vulnerability to rotating wave initiation. We define the vulnerability region, VR, as the range of time delays between conditioning and test waves where the test waves evolves into a rotating wave. The smaller the VR, the more resistant the heart is against origination of dangerous cardiac arrhythmias. Heterogeneity of cardiac muscle is widely recognized as the prerequisite for rotating wave initiation. We have identified the VR in homogeneous 2D excitable media. In the Belousov-Zhabotinsky (BZ) reaction with immobilized catalyst and in the Oregonator model of this reaction, a properly timed test wave gives rise to rotating waves. The VR was increased when the size of the perturbation used for test wave creation was increased or when the threshold for propagation was decreased. Increasing the dimensionality of the medium for 1D to 2D results in diminishing of VR.

Spiral breakup induced by an electric current in a Belousov-Zhabotinsky medium.

Sprial breakup in the Belousov-Zhabotinsky reaction has been observed under the influence of an externally applied alternating electric current. The dynamic mechanism of this breakup is explained in the framework of this reaction. The dependence of the critical electric current amplitude on the period of the wave and on the excitability of the medium is analyzed. Spiral breakup is shown to provide a limit of validity of electric-field-induced drift of vortices in excitable media. Experimental results are complemented with numerical simulations provided by two- and three-variable Oregonator models.

NONLINEAR WAVES, PATTERNS AND SPATIO-TEMPORAL CHAOS IN CELLULAR NEURAL NETWORKS

Spatio-temporal pattern formation occurring in discretely coupled nonlinear dynamical systems has been studied numerically. In this paper, we review the possibilities of using arrays of discretely coupled nonlinear electronic circuits to study these systems. Spiral wave initiation and Turing pattern formation are some of the examples. Sidewall forcing of Turing patterns is shown to be capable of driving the system into a perfect spatial organization, namely, a rhombic pattern, where no defects occur. The dynamics of the two layers supporting Turing and Hopf modes, respectively, is analysed as a function of the coupling strength between them. The competition between these two modes is shown to increase with the diffusion between layers. As well, the coexistence of low- and high-dimensional spatio-temporal chaos is shown to occur in one-dimensional arrays.

Study of reentry initiation in coupled parallel fibers [cardiology]

Initiation of reentries in coupled parallel fibers is studied as a function of several control parameters intrinsic to those fibers. The influence of inhomogeneities in the fibers leading to drift of the vortices and the interaction between them is also analyzed numerically and experimentally. >

Sidewall forcing of hexagonal Turing patterns: rhombic patterns

Abstract Rhombic arrays were obtained by sidewall forcing during Turing pattern formation in numerical simulations. Locking between the frequency of forcing and the wave length between blobs was obtained in accordance with the Farey sequence. This locking appears as a perfect rhombic array oriented in the direction of the imposed forcing. For a constant forcing in duration and amplitude, the following scheme of bifurcation was observed: parallel stripes ↦ rhombic array ↦ domains of hexagons and rhombi separated by “penta-hepta” defects. Symmetry considerations based on a non-uniform stretching along the x-axis were used to describe these transitions. Unstable “varicose-vein” stripes were observed to evolve during the temporal evolution arrays.

Experimental Chua's circuit arrays as an autowave simulator

A two-dimensional (2-D) array of electronic circuits is used to simulate the typical autowaves experimentally found in various active media (target and spiral waves). This experimental setup allows us to obtain some of the phenomena previously described in continuous media, but with a complete control on each cell and on the connections among them.

A geometrical-kinematical approach to spiral wave formation: super-spiral waves

Abstract A geometrical-kinematical approach is used to describe spiral wave behaviour. The dynamic behaviour depends only on the motion of the tip as it follows a pre-determinated trajectory. This tipis considered to be a source for the points making up the spiral wave front. Circular and meandering tip motions are studied in our model. Super-spiral waves are generated when highly meandering tip motion occurs.

General properties of the electric-field-induced vortex drift in excitable media

Abstract The drift of vortices induced by an applied constant electric field is explained by theoretical results obtained from general reaction-diffusion equations. It is shown that the existence of a component of the drift velocity perpendicular to the direction of the external field naturally follows from these equations. General properties are obtained that show the regularity of this phenomenon, independently on the concrete form of the equations. These results confirm the numerical simulations and the experiments performed with an Oregonator model and the Belousov-Zhabotinsky reaction, respectively.

CONTINUOUS CONDUCTIVE VOLUME AFFECTS THE PROPAGATION OF SIGNALS IN DISCRETE SYSTEMS

The effect of a continuous conductive volume on the propagation of signals in discretely coupled nonlinear electronic circuits is presented. Two different coupling mechanisms between the circuits are considered at the same time (through linear resistors and through a continuous conductive volume) trying to mimic the behavior observed in some experiments from cardiac electrophysiology. The effects of the geometry of the conductive medium and the distance from the measuring point to the electronic fiber are studied experimentally.